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Representation Theory

ISSN 1088-4165



$U (\mathfrak{g})$-finite locally analytic representations

Authors: P. Schneider, J. Teitelbaum and Dipendra Prasad
Journal: Represent. Theory 5 (2001), 111-128
MSC (2000): Primary 17B15, 22D12, 22D15, 22D30, 22E50
Published electronically: May 18, 2001
MathSciNet review: 1835001
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In this paper we continue our algebraic approach to the study of locally analytic representations of a $p$-adic Lie group $G$ in vector spaces over a non-Archimedean complete field $K$. We characterize the smooth representations of Langlands theory which are contained in the new category. More generally, we completely determine the structure of the representations on which the universal enveloping algebra $U(\mathfrak g)$ of the Lie algebra $\mathfrak g$of $G$ acts through a finite dimensional quotient. They are direct sums of tensor products of smooth and rational $G$-representations. Finally we analyze the reducible members of the principal series of the group $G=SL_2(\mathbb Q_p)$ in terms of such tensor products.

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  • [B-TVS] Bourbaki, N.: Topological Vector Spaces. Berlin-Heidelberg-New York: Springer-Verlag, 1987. MR 88g:46002
  • [CR] Curtis, C.W., Reiner, I.: Representation theory of finite groups and associative algebras. New York-London: Wiley, 1962. MR 26:2519
  • [DG] Demazure, M., Gabriel, P.: Groupes Algébriques. Amsterdam: North-Holland, 1970. MR 46:1800
  • [Dix] Dixmier, J.: Enveloping Algebras. Revised reprint of 1977 translation, Graduate Studies in Mathematics, 11, Amer. Math. Soc., Providence, R.I., 1996. MR 97c:17010
  • [Fea] Féaux de Lacroix, C. T.: Einige Resultate über die topologischen Darstellungen $p$-adischer Liegruppen auf unendlich dimensionalen Vektorräumen über einem $p$-adischen Körper. Thesis, Köln 1997, Schriftenreihe Math. Inst. Univ. Münster, 3. Serie, Heft 23, pp. 1-111 (1999). MR 2000k:22021
  • [GGP] Gel'fand, I.M., Graev, M.I., Pyatetskii-Shapiro, I.I.: Representation Theory and Automorphic Functions. Academic Press, Boston, 1990. MR 91g:11052
  • [HC] Harish-Chandra: Harmonic Analysis on Reductive $p$-adic Groups. (Notes by G. van Dijk), Lect. Notes Math., vol. 162. Berlin-Heidelberg-New York: Springer-Verlag, 1970. MR 54:2889
  • [Jan] Jantzen, J.C.: Representations of Algebraic Groups. Pure and Applied Mathematics, 131. Academic Press, Boston, 1987. MR 89c:20001
  • [Mor] Morita, Y.: Analytic Representations of $\mathrm{SL}_{2}$ over a $\mathfrak p$-Adic Number Field, III. In Automorphic Forms and Number Theory, Adv. Studies Pure Math. 7, pp. 185-222. North-Holland, Amsterdam, 1985. MR 88b:22019
  • [ST] Schneider, P., Teitelbaum, J.: Locally analytic distributions and $p$-adic representation theory, with applications to $GL_{2}$. Preprint, 1999.
  • [Vig] Vigneras, M.-F.: Représentations $l$-modulaires d'un groupe réductifs $p$-adique avec $l \neq p$. Progress in Math., vol. 137. Birkhäuser Boston, 1996. MR 97g:22007

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Additional Information

P. Schneider
Affiliation: Mathematisches Institut, Westfälische Wilhelms-Universität Münster, Einsteinstr. 62, D-48149 Münster, Germany

J. Teitelbaum
Affiliation: Department of Mathematics, Statistics, and Computer Science (M/C 249), University of Illinois at Chicago, 851 S. Morgan St., Chicago, Illinois 60607

Dipendra Prasad
Affiliation: Harish-Chandra Research Institute, Chhatnag Road, Jhusi, Allahabad, 211019, India

Received by editor(s): August 2, 2000
Received by editor(s) in revised form: September 25, 2000
Published electronically: May 18, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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